4 Dark Energy in f (R) Theories

In this section we apply f (R) theories to dark energy. Our interest is to construct viable f (R) models that
can realize the sequence of radiation, matter, and accelerated epochs. In this section we do not attempt to
find unified models of inflation and dark energy based on f (R) theories.

Originally the model () was proposed to explain the late-time cosmic
acceleration [113, 120, 114, 143] (see also [456, 559, 17, 223, 212, 16, 137, 62] for related works).
However, this model suffers from a number of problems such as matter instability [215, 244], the instability
of cosmological perturbations [146, 74, 544, 526, 251], the absence of the matter era [28, 29, 239], and
the inability to satisfy local gravity constraints [469, 470, 245, 233, 154, 448, 134]. The main reason why
this model does not work is that the quantity is negative. As we will see later, the
violation of the condition gives rise to the negative mass squared for the scalaron field.
Hence we require that to avoid a tachyonic instability. The condition is also
required to avoid the appearance of ghosts (see Section 7.4). Thus viable f (R) dark energy models need to
satisfy [568]

where is the Ricci scalar today.

In the following we shall derive other conditions for the cosmological viability of f (R) models. This
is based on the analysis of [26]. For the matter Lagrangian in Eq. (2.1) we take into
account non-relativistic matter and radiation, whose energy densities and satisfy